x^2-nx-1=0
n=1
x^2-x-1=0
x^2-x=1
x^2-x+1/4=1+1/4
(x-1/2)^2=4/4+1/4
(x-1/2)^2=5/4
x-1/2=√5/2
x=1/2+√5/2
x=(1+√5)/2
x=(1+√5)/2=1.6=φ
φ=1.6
x^2+x-1=0
x^2+x=1
x^2+x+1/4=1+1/4
(x+1/2)^2=4/4+1/4
(x+1/2)^2=5/4
x+1/2=√5/2
x=-1/2+√5/2
x=(-1+√5)/2
x=(-1+√5)/2=0.6=1/φ
1/φ=0.6

Fibonacci
Fibonacci Numbers
The Fibonacci numbers
Fibonacci Sequences
The Fibonacci sequences

フィボナッチ数
フィボナッチ数列

An+2=An+An+1
Fn+2=Fn+Fn+1
F0=0
F1=1
An=(1/√5){[(1+√5)/2]^2-[(1-√5)/2]^2}
An=[φ^n-(1-φ)^n]/√5
An=[φ^n-(-φ)^-n]/√5
Fn=(1/√5){[(1+√5)/2]^2-[(1-√5)/2]^2}
Fn=[φ^n-(1-φ)^n]/√5
Fn=[φ^n-(-φ)^-n]/√5

An/An-1=r=φ=1.6
An-1/An=1/r=1/φ=0.6
Fn/Fn-1=r=φ=1.6
Fn-1/Fn=1/r=1/φ=0.6

An+1/An=r=φ=1.6
An+1/An=1/r=1/φ=0.6
Fn+1/Fn=r=φ=1.6
Fn+1/Fn=1/r=1/φ=0.6

Geometric Progression
The geometric progressions
Geometric Sequence
The geometric sequences
等比数列
とうひすうれつ
Recurrence Relation
The recurrence relations
漸化式
ぜんかしき

(1/φ)φ=1
φ=1.6
(1/1.6)1.6=(0.6)(1.6)
(0.6)(1.6)=0.96
φn=(Fn)φ+Fn-1

Fn+1/Fn=1.618

Fn+2/Fn=2.618

Fn/Fn+2=0.382

黄金比
おうごんひ
Golden Triangle
The golden triangles
Goldenes Dreieck
Triangle d'or
黄金三角形
おうごんさんかっけい
Golden Rectangle
The golden rectangles
Goldenes Rechteck
黄金長方形
おうごんちょうほうけい
Pentagrammum
Pentagram
The pentagrams
Pentagramm
五芒星
ごぼうせい
Pentagonum
Pentagon
The pentagons
Fünfeck
正五角形
せいごかっけい
Regular Dodecahedron
正十二面体
せいじゅうにめんたい

体温

x^1=1x
x^2=1x+1
x^3=2x+1
x^4=3x+2
x^5=5x+3
x^6=8x+5
x^7=13x+8
x^8=21x+13
x^9=34x+21


x^2+nx-1=0
n=2
x^2+2x-1=0
x^2+2x=1
x^2+2x+1=1+1
x^2+2x+1=2
(x+1)^2=2
x+1=√2
x=-1+√2
√2≒1.4
x≒-1+1.4
x≒0.4
x^2-2x-1=0
x^2-2x=1
x^2-2x+1=1+1
x^2-2x+1=2
(x-1)^2=2
x-1=√2
x=1+√2
√2≒1.4
x≒1+1.4
x≒2.4

x^2+nx-1=0
n=3
x^2+3x-1=0
x^2+3x=1
x^2+3x+(3/2)^2=1+(3/2)^2
x^2+3x+9/4=1+9/4
(x+3/2)^2=4/4+9/4
(x+3/2)^2=13/4
x+3/2=√13/2
x=-3/2+√13/2
x=(-3+√13)/2
√13≒3.6
x≒(-3+3.6)/2
x≒0.6/2
x≒0.3
x^2-3x-1=0
x^2-3x=1
x^2-3x+(3/2)^2=1+(3/2)^2
x^2-3x+9/4=1+9/4
(x-3/2)^2=4/4+9/4
(x-3/2)^2=13/4
x-3/2=√13/2
x=3/2+√13/2
x=(3+√13)/2
√13≒3.6
x≒(3+3.6)/2
x≒6.6/2
x≒3.3